Scilab Function
Last update : 6/7/2004

interp - cubic spline evaluation function

### Calling Sequence

  `[yp [,yp1 [,yp2 [,yp3]]]]=interp(xp, x, y, d [, out_mode])`

### Parameters

- `xp`: real vector or matrix
- `x,y,d`: real vectors of the same size defining a cubic spline or sub-spline function (called `s` in the following)
- `out_mode`: (optionnal) string defining the evaluation of `s` outside the [x1,xn] interval
- `yp`: vector or matrix of same size than `xp`, elementwise evaluation of `s` on `xp` (yp(i)=s(xp(i) or yp(i,j)=s(xp(i,j))
- `yp1, yp2, yp3`: vectors (or matrices) of same size than `xp`, elementwise evaluation of the successive derivatives of `s` on `xp`

### Description

Given three vectors `(x,y,d)` defining a spline or sub-spline function (see `splin`) with `yi=s(xi), di = s'(xi)` this function evaluates `s` (and `s', s'', s'''` if needed) at `xp(i)` :

          yp(i) = s(xp(i))    or  yp(i,j) = s(xp(i,j))
          yp1(i) = s'(xp(i))   or  yp1(i,j) = s'(xp(i,j))
          yp2(i) = s''(xp(i))   or  yp2(i,j) = s''(xp(i,j))
          yp3(i) = s'''(xp(i))   or  yp3(i,j) = s'''(xp(i,j))

The `out_mode` parameter set the evaluation rule for extrapolation, i.e. for `xp(i)` not in [x1,xn] :

  - "by_zero": an extrapolation by zero is done
  - "by_nan": extrapolation by Nan
  - "C0": the extrapolation is defined as follows :

              s(x) = y1  for  x < x1
              s(x) = yn  for  x > xn

  - "natural": the extrapolation is defined as follows (p_i being the polynomial defining `s` on [x_i,x_{i+1}]) :

              s(x) = p_1(x)      for  x < x1
              s(x) = p_{n-1}(x)  for  x > xn

  - "linear": the extrapolation is defined as follows :

              s(x) = y1 + s'(x1)(x-x1)  for  x < x1
              s(x) = yn + s'(xn)(x-xn)  for  x > xn

  - "periodic": `s` is extended by periodicity.

### Examples

    // see the examples of splin and lsq_splin

    // an example showing C2 and C1 continuity of spline and subspline
    a = -8; b = 8;
    x = linspace(a,b,20)';
    y = sinc(x);
    dk = splin(x,y);  // not_a_knot
    df = splin(x,y, "fast");
    xx = linspace(a,b,800)';
    [yyk, yy1k, yy2k] = interp(xx, x, y, dk);
    [yyf, yy1f, yy2f] = interp(xx, x, y, df);
    xbasc()
    subplot(3,1,1)
    plot2d(xx, [yyk yyf])
    plot2d(x, y, style=-9)
    legends(["not_a_knot spline","fast sub-spline","interpolation points"],...
            [1 2 -9], "ur",%f)
    xtitle("spline interpolation")
    subplot(3,1,2)
    plot2d(xx, [yy1k yy1f])
    legends(["not_a_knot spline","fast sub-spline"], [1 2], "ur",%f)
    xtitle("spline interpolation (derivatives)")
    subplot(3,1,3)
    plot2d(xx, [yy2k yy2f])
    legends(["not_a_knot spline","fast sub-spline"], [1 2], "lr",%f)
    xtitle("spline interpolation (second derivatives)")

    // here is an example showing the different extrapolation possibilities
    x = linspace(0,1,11)';
    y = cosh(x-0.5);
    d = splin(x,y);
    xx = linspace(-0.5,1.5,401)';
    yy0 = interp(xx,x,y,d,"C0");
    yy1 = interp(xx,x,y,d,"linear");
    yy2 = interp(xx,x,y,d,"natural");
    yy3 = interp(xx,x,y,d,"periodic");
    xbasc()
    plot2d(xx,[yy0 yy1 yy2 yy3],style=2:5,frameflag=2,leg="C0@linear@natural@periodic")
    xtitle(" different way to evaluate a spline outside its domain")

### See Also

`splin`, `lsq_splin`,

### Author

B. Pincon